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30.7 Pure radiation fields 469For M =0, and with the ansatz P = P (u, B), one gets (Stephani1980)4σ 2 B 2 (P ,BB + P ,uu )+4σ(2σ − 3)BP ,B +(σ − 2)(σ − 3)P =0. (30.74)Solutions can be found by standard separation methods or by takingP = B ν f(s), s= u/B, which leads to the hypergeometric equation4σ 2 (s 2 +1)f ′′ +4σ(3 − 2σν)sf ′+[4σν(σν + σ − 3)+(σ − 2)(σ − 3)] f =0.(30.75)This solution generalizes the Hauser solution (29.72), which is containedfor m =0,σ=2,ν=3/4. A large class of solutions to (30.74) has beengiven by Tafel et al. (1991), and solutions with M = 0 admitting a Killingvector i(∂ ζ − ∂ ζ) by Lewandowski et al. (1991).The third case is that all metric functions are independent of u. Theyhave to satisfy m +iM = const, L =iB(ζ,ζ) ,ζ andP 2 ( P 2 B ,ζζ),ζζ +2P 4 B ,ζζ(ln P ) ,ζζ= M. (30.76)Solutions which admit at least an H 3 of homothetic motions or haveM = 0 were found by Lewandowskiand Nurowski(1990) and Grundlandand Tafel (1993) (note that solutions 3.15 and 3.17 of Grundland andTafel (1993) contain mistakes).